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Shot noise in magnetic field modulated graphene superlattice
Author’s Accepted Manuscript Shot noise in magnetic field modulated graphene superlattice Farhad Sattari
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S1386-9477(15)30033-3 http://dx.doi.org/10.1016/j.physe.2015.04.025 PHYSE11948
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 20 February 2015 Revised date: 22 April 2015 Accepted date: 28 April 2015 Cite this article as: Farhad Sattari, Shot noise in magnetic field modulated graphene superlattice, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.04.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Shot noise in magnetic field modulated graphene superlattice Farhad Sattari*
Department of Physics, Faculty of Sciences, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran Abstract We investigate the shot noise properties in a monolayer graphene superlattice modulated by N parallel ferromagnets deposited on a dielectric layer. It is found that for the antiparallel magnetization configuration or when magnetic field is zero the new Dirac-like point appears in graphene superlattice. The transport is almost forbidden at this new Dirac-like point, and the Fano factor reaches its maximum value 1/3. In the parallel magnetization configuration as the number of magnetic barriers increases, the shot noise increases. In this case, the transmission can be blocked by the magnetic-electric barrier and the Fano factor approaches 1, which is dramatically distinguishable from that in antiparallel alignment. The results may be helpful to control the electron transport in graphene-based electronic devices.
Keywords Graphene superlattice; Shot noise; Fano factor; Dirac point.
1. Introduction Graphene, two-dimensional material tightly packed into a monolayer honeycomb lattice of carbon atoms, which was synthesized by Novoselov et al. in 2004 [1,2]. In low energy regime, the quasiparticles in graphene close to the Dirac points (often referred to as K and
K ) are described by the massless Dirac-like equation. Such a peculiar band structure results in many unique electronic properties, including the unconventional quantum Hall effect [3], *
Tel.: +98 45 33514702; fax: +98 45 33514701 E-mail address: [email protected] (F. Sattari)
1
the nonzero conductivity at vanishing carrier concentration [4,5], the reflectionless Klein tunneling [6], the sub-Poissonian shot noise [7,8], special Andreev reflection [9], and many others. In 1970 the superlattice was proposed by Esaki and Tsu [10], which was attracted a great deal of researches over the past decades years on the transport properties of the superlattice [1114]. Motivated by the experimental realization of graphene superlattice [15-17], electronic bandgap structures and transport properties in graphene superlattice with electrostatic potential barrier was extensively investigated [18-23]. The transport properties in graphenebased superlattice structure were first studied by Bai and Zhang [18] the authors found that the conductivity of the graphene superlattice depends on the superlattice structural parameters. The conductance of a disordered graphene superlattice was investigated in Ref. [19], and the authors found that the conductance vanishes when the sample size becomes very large. In Ref. [20], the spin transport properties of graphene superlattice in the presence of Rashba spin-orbit interaction was studied and found that the magnetoresistance ratio shows a strong dependence on the number of barriers. To circumvent the Klein tunneling effect, it was suggested that a magnetic barrier can effectively block Klein tunneling and achieve confinement for massless Dirac fermions in graphene [24]. The required magnetic structures in graphene can be realized by depositing ferromagnetic stripes on the graphene layer [25,26]. There also exist many theoretical works which were studied transport properties through magnetic barriers and magnetic superlattice in graphene [27-34]. Shot noise originates from the fluctuation in the electrical signal due to the discreteness of electron charges. In addition, it is well known that shot noise is useful to obtain information of transmission that is not available through the standard conductance measurements [35,36]. The shot noise is characterized by the Fano factor F being the ratio of the noise power to mean current [37]. Recently, some papers focus on the shot noise in semiconductor
2
superlattices and graphene based nano-structures both theoretically and experimentally [7,8,38-51]. Cheianov and Falko studied the unusual Fano factor in the graphene n–p junctions [38]. Tworzydlo et al. predicted that the Fano factor for a wide and short sample has a maximum value of 1/3 at the Dirac point, which is 1/3 of Poissonian value [39]. This is the same value as in the diffusive metals. Zhu and Guo investigated shot noise in the graphenebased double barriers and found that the shot noise with the Fano factor equal to 1/3 occurs at the Dirac point [40]. The transport properties and shot noise in Thue-Morse sequence graphene superlattice were investigated by Huang et al. [42] the authors found that the Fano factor has a maximum close to 1/3 in the vicinity of Dirac point. Also, experimental results of shot noise measurement in graphene structures are in good agreement with the theoretical predication [7,52]. It is generally accepted that the sub-Poisson shot noise of graphene originated from the peculiar band structure at the low energy regime near the Dirac point. In other words, the quasiparticles in graphene are described by the massless Dirac-like equation rather than the Schröedinger equation. The purpose of this paper is to study the shot noise in magnetic field modulated graphene superlattice by using the transfer matrix method. The effect of the number of barriers on the Fano factor is taken into account. We show that for the antiparallel magnetization configuration or when magnetic field is zero the Fano factor reaches its maximum value 1/3, whereas for the parallel magnetization configuration it can be approached to 1 due to the transmission blockage. In particular, our probe shows that for the parallel magnetization and when the number of barriers, N, are larger than 50 the Fano factor reaches a Poissonian value, which is dramatically distinguishable from the semiconductor superlattice. In the semiconductor superlattice when all the barriers are identical, in the N → ∞ limit the shot noise approach 1/3 of the Poissonian value [50,51]. The rest of the paper is organized as
3
follows; our method and formalism are described in the next section. In Section 3 we present and discuss our results, and finally we end the paper with a brief conclusion.
2. Model and theory In this paper, we consider two kinds of systems. In both cases, a monolayer graphene covered by a thin insulating layer and parallel ferromagnetic (FM) stripes are deposited on the top of the dielectric layer [25,26]. Further, a gate voltage is applied to the FM stripes in order to produce an electrostatic barrier. In both cases, FM stripes have magnetization perpendicular to the graphene in the x-y plane. In the first and the second cases FM stripes with magnetization parallel (P) or antiparallel (AP) perpendicular to the graphene in the x-y plane are deposited on top of the dielectric layer, respectively. As known, the magnetic field generated by such a strip is non-uniform and depends on both its size and the distance to the studied 2D system. However, for sufficiently thin ferromagnetic strip, it is almost constant under the strip and vanishes outside this region [53]. Thus, the systems under consideration are graphene superlattice modulated by magnetic field. The magnetic field B(x) emerging from the FM stripes will influence locally the motion of Dirac electrons in the graphene x-y plane and is assumed to be homogeneous in the y direction, but varies along the x direction. The schematic of the structures is shown in Fig. 1. According to the above discussion, the potential profile of the system consists of a sequence of N electrostatic barriers of equal height U 0 and width b, modulated by N magnetic barriers. Where, electrostatic and magnetic barriers separated by well regions (nonmagnetic regions) of width w. Here, the effects of electron-electron and electron-phonon interactions are neglected by considering a single electron transmission at zero temperature. Therefore, the charge carriers in our model are described by the following Hamiltonian
Hˆ Hˆ 0 V ( x) Iˆ,
(1)
4
in which, ˆ v σ H .( p eA) , 0 F
(2)
U V ( x) 0 0
(3)
in barrier, in well,
, where p ( px , px ) is the quasiparticle momentum (setting =1), σ ( x , x ) is the 2D Pauli matrix, vF 106 ms 1 is the Fermi velocity, Iˆ is a 2×2 unit matrix, and A =[0, Ay (x) ,
0] is the vector potential with x A( x) B( x) in the Laudau gauge. For simplicity, we express all the quantities in the dimensionless form by rescaling: the magnetic field B( x) B0 B( x), the vector potential
A( x) B0l B A( x) with the magnetic length l B / eB0 , the
wavevector k k / lB , the electron energy E EvF / lB , the barrier height U 0 U 0 vF / lB , the barrier of width b bl B and well of with w wl B . For a realistic value B0 0.1 T [54,55] we find l B 80 nm and vF / lB 7 meV, which set the typical length and energy scales. To solve Eq. (1), we suppose that an incident electron from the left will go towards the interface with incident angle and energy E. Moreover, for the sake of simplicity we divide barrier regions into M (M 1) segments, each of which has width a =b/M. The vector potential in each segment can be considered as constant and then the plane-wave
ˆ can be approximation can be taken [56]. The general solution to the Hamiltonian H expressed in the following form
b e
ik y y
1 a e iqx x q x ik y i E U0
1 iq x x q ik y bi e x E U0
,
1 d e ik x x k x ik y i E
,
(4)
in barriers and
w e
ik y y
1 ci e ik x x k x ik y E
5
(5)
in wells. Where E is the energy of incident electron, q x ( E U 0 ) 2 k y2 and k x E cos are the wave vectors along the x direction in the barrier and well regions respectively, while k y E sin A( x) is the wave vectors along the y axis. Also a, b and c, d represent the
ampliudes of quasiparticles in the barrier and well regions, respectively. By appling the continuity of wave functions at the boundaries for a system consisting of N barriers and using the transfer-matrix method, we obtain t and r which represent the reflection and transmission coefficients, respectively. Then transmission probability T and reflection probability R can be evaluated by T P ( AP ) t P ( AP )
2
and R P ( AP ) r P ( AP )
2
respectively. Here, P
and AP denote the parallel and antiparallel alignments, respectively. Thus, the zero temperature conductivity through the magnetic field modulated graphene superlattice can be calculated by means of Büttiker formula [57] /2
G P ( AP ) G0
T
P ( Ap )
( ) cos( )d ,
(6)
/2
where G0 2e 2 EF Ly / vF 2 in which L y denotes the width of the graphene strip along the y direction. A convenient measure of shot noise is the Fano factor, and it was defined as the ratio of noise power and mean current and given by [39] /2
F P ( AP )
T
/2
P ( Ap )
(1 T P ( AP ) )( ) cos( )d
T
(7)
.
/2
P ( AP )
( ) cos( )d
/2
According to the Eq. (7) the Fano factor F=0 in the perfect transmission case and F=1 when the transmission is blocked.
3. Numerical result and discussion
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In this section we investigate the transmission probability and shot noise in the magnetic field modulated monolayer graphene superlattice by using the methods described in the previous section. In all the calculations, energy E of the incident electrons, barrier height U 0 , well width w, barrier width b and the magnetic field B are taken to be 5, 10, 0.2, 0.2 and 1 respectively, otherwise it is specified. At first, we calculate the transmission probability as a function of barrier height and incident angle through the system with different number of barriers when magnetic field is zero. The results are shown in Fig. 2. As it is clear from Fig. 2 transmission probability is very small at the U 0 E (1 w / b) . Thus, this point is new Dirac-like point. This phenomenon is very clear when the number of barriers is large enough. The new Dirac-like point is determined by the ratio of w/b. By using the transfer-matrix method, one can find the expression for the transmission probability T ( ) t at the new Dirac-like point as
t
(2 cos ) 2 N 1 , i ( 2 k xl2 N ) 2 1 4[ M 11 sin i sin ( M 12e M 21e i ) ) M 22ei 2 k xl2 N ]
M 11 M 12 2 N 1 2i sin i 2ik l M 21 M 22 i 2 2e e x i
2e i e 2ik xli , 2i sin
(8)
(9)
where li is the length of system at ith boundary. In addition, due to the interface of the well and the barrier, the transmission probability oscillates with the variation of the potential barriers height and by increasing the number of barriers more resonant peaks (valleys) appear in the transmission probability at some certain U 0 . The resonance condition is given by a function that yields f (q x , w, b) . For instance, for
the case N=1, the condition yields q x b m (m is an integer). For N 2 when we have the condition that (q x k x ) (b w) 2m , the transmission has finite values at angles different from 0 . This is due to the resonance process in a system with N barriers [19]. Fig. 3 shows the Fano factor with different number of barriers when magnetic field is zero. In the 7
vicinity of the Dirac-like point, the Fano factor shows a peak with value approximately F=1/3 which is originating from quasiparticle chirality and Klein tunneling. Meanwhile, in the region far from this point it oscillates with decreasing amplitude. This is a result of the completely ballistic tunneling, since the classical dynamics of the Dirac fermions is ballistic [39]. The Fano factor is sensitive to the ratio of w/b, when the well and the barrier width are changed simultaneously, the Fano factor shows the robust properties, because the position of the new Dirac point does not shift. With increasing the well width, the peak of the Fano factor shift right. It should be noted that if we choose all the parameters as Ref. [58] then figure 3 would be similar to figure 3 in Ref. [58]. With the condition that the magnetic field is applied and the antiparallel magnetization configuration, the Fano factor was calculated and the results was plotted in Fig. 4. As it is clear in Fig. 4, the Fano factor is a maximum F = 1/3 near the U 0 E (1 w / b) , while it oscillates with smaller amplitude in the region far from this point. This is similar to the zero magnetic field case, as a result when a monolayer graphene superlattice modulated by N FM stripes with antiparallel magnetization the position of the new Dirac point is robust against the magnetic field. It is necessary to mention that in Fig. 4, B=1 and the other parameters in Figs. 4(a), (b) and (c) are the same as in Figs. 3(a), (b) and (c), respectively. Fig. 5 presents the Fano factor in magnetic field modulated monolayer graphene superlattice. In this figure FM stripes with magnetization parallel is perpendicular to the graphene layer. As it is obvious in Fig. 5 the Fano factor increases by increasing the number of barriers. We note that when the number of barriers is big enough unlike antiparallel magnetization structure the Fano factor reaches a Poissonian value. Because the transmission can be blocked by the magnetic-electric barrier provided by the ferromagnets. These features can be intuitively conceived from the following discussion. The emergence angle f at right is obtained by exploiting conservation of wave vector in y direction 8
(8)
(10)
sin f sin Nb / E.
Eq. (10) implies that transmission through the structure is only possible when satisfied the following condition 1 sin
Nb 1. E
(11)
Another explanation for this phenomenon is that when sin ( Nb / E ) 1 the electron wave vector in the 2Nth strip is imaginary, which corresponds to an evanescent wave. Thus the transmission probability decreases by increasing N and/or decreasing E. In other words, by controlling the ratio Nb / E one can control the Fano factor and this renders the structure’s efficient wave-vector filter for Dirac electrons. It is observed from Fig. 5c that a striking Poisson value plateau of the Fano factor is formed for any barrier height. Because for N 50 the sin ( Nb / E ) is always more than 1, which means for any barrier height the Fano factor is 1. To further understand the variation of the Fano factor in magnetic field modulated graphene superlattice, we calculated the Fano factor as a function of the number of barriers with w=b=0.2. The result is plotted in Fig. 6. We can see that at U 0 2 E for the antiparallel magnetization configuration or when magnetic field is zero, the Fano factor is far below 1/3. Moreover, at U 0 2 E and for one barrier structure the Fano factor is far away from 1/3. By increasing the number of barriers, a periodical structure is formed, the point U 0 2 E becomes a Dirac-like point, and the Fano factor in this point approaches 1/3. Meanwhile, from Fig. 6, we find that in the parallel magnetization configuration as the number of magnetic barriers increases, the shot noise increases. It is also very clear when the number of barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value for any barrier height due to the transmission blockage.
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4. Conclusion Based on the transfer-matrix method, we have investigated the shot noise in magnetic field modulated graphene superlattice. We have shown that for the antiparallel magnetization configuration or in absence of the magnetic field the Fano factor shows a peak at the new Dirac point with value approximately F=1/3. This is due to massless Dirac fermions and directly related to Klein tunneling. In other words, at the new Dirac point the transport is almost prohibited, as a result the conductivity gets the minimum value and the Fano factor reaches the maximum value. Meanwhile, for the antiparallel magnetization the position of the new Dirac point is robust against the magnetic field. Besides, for the parallel magnetization the Fano factor increases by increasing the number of barriers. When the number of barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value due to the transmission blocked by the magnetic-electric barrier provided by the ferromagnets.
References [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Griegorieva, A. A. Firsov, Science 306 (2004) 666. [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Griegorieva, A. A. Firsov Nature (London) 438 (2005) 197. [3] Y. Zhang, Y. Wen Tan, H. L. Stormer, P. Kim, Nature 438 (2005) 201. [4] K. Ziegler, Phys. Rev. B 75 (2007) 233407. [5] M. I. Katsnelson, Eur. Phys. J. B 51 (2006) 157. [6] M. I. Katsnelson, K. S. Nososelov, A. K. Geim, Nat. phys. 2 (2006) 620. [7] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, P. J. Hakonen, Phys. Rev. Lett. 100 (2008) 196802.
10
[8] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Mor purgo, P. J. Hakonen, J. Low Temp. Phys. 153 (2008) 374. [9] C. W. J. Beenakker, Phys. Rev. Lett. 97 (2006) 067007. [10] L. Esaki, R. Tsu and, IBM J. Res. Develop. 14 (1970) 61. [11] F. Z. Meghoufel, S. Bentata, S. Terkhi, F. Bendahma, S. Cherid, Superlattices Microstruct 57 (2013) 115. [12] G. Bastard, E. E. Mendez, L. L. Chang, L. Esak, Phys. Rev. B 28 (1983) 3241. [13] A. Sibille, J. F. Palmier, h. Wang, F. Mollot, Phys. Rev. Lett. 64 (1990) 52. [14] James Leo, Phys. Rev. B 39 (1989) 5947. [15] J. C. Meyer, C. O. Girit, M. F. Crommie, A. Zettl, Appl. Phys. Lett. 92 (2008) 123110. [16] S. Marchini, S. Günther, J. Wintterlin, Phys. Rev. B 76 (2007) 075429. [17] A. L. Vazquez de Parga, F. Calleja, B. Borca, M. C. G. Passeggi, Jr., J. J. Hinarejos, F. Guinea, R. Miranda, Phys. Rev. Lett. 100 (2008) 056807. [18] C. Bai, X. Zhang, Phys. Rev. B 76 (2007) 075430. [19] N. Abedpour, A. Esmailpour, R. Asgari, M. R. Rahimi Tabar, Phys. Rev. B 79 (2009) 165412. [20] F. Sattari, E. Faizabadi, Solid State Commun 79 (2014) 48. [21] E. Faizabadi, M. Esmaeilzadeh, F. Sattari, Eur. Phys. J. B 85 (2012) 30073. [22] L. Brey, H. A. Fertig, Phys. Rev. Lett. 103 (2009) 046809. [23] C. H. Park, L. Yang, Y. W. Son, M. L. Cohen, S. G. Louie, Phys. Rev. Lett. 101 (2008) 126804. [24] A. De Martino, L. Dell’Anna, R. Egger, Phys. Rev. Lett. 98 (2007) 066802. [25] B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, D. Goldhaber-Gordon, Phys. Rev. Lett. 98 (2007) 236803. [26] J. R. Williams, L. Di Carlo, C. M. Marcus, Science 317 (2007) 638. [27] L. Dell’Anna, A. De Martino, Phys. Rev. B 79 (2009) 045420. 11
[28] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, J. Phys.: Condens. Matter 22 (2009) 465302. [29] L. Z. Tan, C.-H. Park, S. G. Louie, Phys. Rev. B 81 (2010) 195426. [30] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, Appl. Phys. Lett. 93 (2008) 242103. [31] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, New J. Phys. 11 (2009) 095009. [32] Q. Sh. Wu, Sh. N. Zhang, Sh. J. Yang, J. Phys.: Condens. Matter 20 (2008) 485210. [33] Q. H. Huo, R. Z. Wang, H. Yan, Appl. Phys. Lett. 101 (2012) 152404. [34] F. Sattari, E. Faizabadi, Eur. Phys. J. B 86 (2013) 40275. [35] C. W. J. Beenakker, Phys. Rev. B 46 (1992) 1889. [36] J. C. Egues, G. Burkard, D. S. Saraga, J. Schliemann, D. Loss, Phys. Rev. B 72 (2005) 235326. [37] Y. M. Blanter, M. Büttiker, Phys. Rep. 336 (2000) 1. [38] V. V. Cheianov, V. I. Falko, Phys. Rev. B 74 (2006) 041403. [39] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, C. W. J. Beenakker, Phys. Rev. Lett. 96 (2006) 246802. [40] R. Zhu, Y. Guo, Appl. Phys. Lett. 91 (2007) 252113. [41] Y .Gong, Y.Guo, J. Appl. Phys. 106 (2009) 064317. [42] H. Huang, D. Liu, H. Zhang, X. Kong, J. Appl. Phys. 113 (2013) 043702. [43] E. B. Sonin Phys. Rev. B 77 (2008) 233408. [44] Y-Xian Li, L.-Fei. Xu, Solid State Commun 151 (2011) 219. [45] E. S. Azarova, G. M. Maksimova, Physica E 61 (2014) 118. [46] Q. P. Wu, X. D. He, Z. F. Liu Eur. Phys. J. B 85 (2012) 346. [47] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, P. J. Hakonen, Solid State Commun 149 (2009) 1050. [48] S. Wu, Y.Guo, Physica E 59 (2014) 158. [49] B. H. Wu, J.C. Cao, Physica E 40 (2008)1319. 12
[50] M. J. M. de Jong, C. W. J. Beenakker, Phys. Rev. B 51 (1995) 16867. [51] W. Song, A. K. M. Newaz, J. K. Son, E. E. Mendez, Phys. Rev. Lett. 96 (2006) 126803. [52] L. DiCarlo, J. R. Williams, Y. Zhang, D. T. McClure, C. M. Marcus, Phys. Rev. Lett. 100 (2008) 156801. [53] A. Matulis, F. M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 1518. [54] F. Zhai, K. Chang, Phys. Rev. B 77 (2008) 113409. [55] A. T. Ngo, J. M. Villas-Boas, S. E. Ulloa, Phys. Rev. B 78 (2008) 245310. [56] Z. Y. Zeng, L. D. Zhang, X. H. Yan, J. Q. You, Phys. Rev. B 60 (1999) 1515. [57] M. Büttiker, Phys. Rev. Lett. 57 (1986) 1761. [58] X. X. Guo, D. Liu, Y. X. Li, Appl. Phys. Lett. 98 (2011) 242101.
Figure captions: Fig. 1. (a) Schematic diagram of the model, the monolayer graphene covered by a thin insulating layer, parallel FM stripes are deposited on top of the insulating layer. Each FM stripe has a rectangular cross section and a magnetization parallel to the z axis. The gate voltage V g applied on the FM stripes induces potential barrier in the graphene sheet. The FM stripes have magnetization parallel (P) or antiparallel (AP) to the z axis. (b) Magnetic field profile B(x) (red dashed line), corresponding vector potential A(x) (blue solid line) and the electrostatic potential U 0 (green dotted line) for the P alignment. (c) The same as in (b) but for the AP alignment.
Fig. 2. The transmission probability as a function of barrier height and incident angle for electrons traversing the considered structure. The number of barriers is (a) five, (b) twenty and [(c) and (d)] fifty. The well width is [(a), (b) and (c)] 0.2 and (d) 0.3. Fig. 3. The Fano factor as a function of barrier height for graphene superlattice with different the number of barriers in absence of the magnetic field. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
13
Fig. 4. The Fano factor as a function of barrier height for graphene superlattice modulated by N FM stripes with antiparallel magnetization. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
Fig. 5. The Fano factor as a function of barrier height for graphene superlattice modulated by N FM stripes with parallel magnetization configuration. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
Fig. 6. The Fano factor as a function of the number of barriers for graphene superlattice modulated by N FM stripes with antiparallel configuration magnetization or when magnetic field is zero (green dashed and blue solid line) and parallel magnetization configuration (red dotted and black dashed-dotted line). Green dashed and black dashed-dotted line correspond to U 0 2E and blue solid and red dotted line correspond to U 0 2E .
Fig. 1.
Fig. 2.
14
Fig. 3.
15
Fig. 4.
Fig. 5.
16
Fig. 6.
Graphical abstract In the parallel magnetization configuration when the number of the barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value for any barrier height.
Highlights We investigate the shot noise properties through graphene superlattice.
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The Fano factor depends on the number of barriers. For the parallel configuration the Fano factor reaches a Poissonian value. For the antiparallel configuration the Fano factor at the Dirac point equals 1/3.
18
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PII: DOI: Reference:
S1386-9477(15)30033-3 http://dx.doi.org/10.1016/j.physe.2015.04.025 PHYSE11948
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 20 February 2015 Revised date: 22 April 2015 Accepted date: 28 April 2015 Cite this article as: Farhad Sattari, Shot noise in magnetic field modulated graphene superlattice, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.04.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Shot noise in magnetic field modulated graphene superlattice Farhad Sattari*
Department of Physics, Faculty of Sciences, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran Abstract We investigate the shot noise properties in a monolayer graphene superlattice modulated by N parallel ferromagnets deposited on a dielectric layer. It is found that for the antiparallel magnetization configuration or when magnetic field is zero the new Dirac-like point appears in graphene superlattice. The transport is almost forbidden at this new Dirac-like point, and the Fano factor reaches its maximum value 1/3. In the parallel magnetization configuration as the number of magnetic barriers increases, the shot noise increases. In this case, the transmission can be blocked by the magnetic-electric barrier and the Fano factor approaches 1, which is dramatically distinguishable from that in antiparallel alignment. The results may be helpful to control the electron transport in graphene-based electronic devices.
Keywords Graphene superlattice; Shot noise; Fano factor; Dirac point.
1. Introduction Graphene, two-dimensional material tightly packed into a monolayer honeycomb lattice of carbon atoms, which was synthesized by Novoselov et al. in 2004 [1,2]. In low energy regime, the quasiparticles in graphene close to the Dirac points (often referred to as K and
K ) are described by the massless Dirac-like equation. Such a peculiar band structure results in many unique electronic properties, including the unconventional quantum Hall effect [3], *
Tel.: +98 45 33514702; fax: +98 45 33514701 E-mail address: [email protected] (F. Sattari)
1
the nonzero conductivity at vanishing carrier concentration [4,5], the reflectionless Klein tunneling [6], the sub-Poissonian shot noise [7,8], special Andreev reflection [9], and many others. In 1970 the superlattice was proposed by Esaki and Tsu [10], which was attracted a great deal of researches over the past decades years on the transport properties of the superlattice [1114]. Motivated by the experimental realization of graphene superlattice [15-17], electronic bandgap structures and transport properties in graphene superlattice with electrostatic potential barrier was extensively investigated [18-23]. The transport properties in graphenebased superlattice structure were first studied by Bai and Zhang [18] the authors found that the conductivity of the graphene superlattice depends on the superlattice structural parameters. The conductance of a disordered graphene superlattice was investigated in Ref. [19], and the authors found that the conductance vanishes when the sample size becomes very large. In Ref. [20], the spin transport properties of graphene superlattice in the presence of Rashba spin-orbit interaction was studied and found that the magnetoresistance ratio shows a strong dependence on the number of barriers. To circumvent the Klein tunneling effect, it was suggested that a magnetic barrier can effectively block Klein tunneling and achieve confinement for massless Dirac fermions in graphene [24]. The required magnetic structures in graphene can be realized by depositing ferromagnetic stripes on the graphene layer [25,26]. There also exist many theoretical works which were studied transport properties through magnetic barriers and magnetic superlattice in graphene [27-34]. Shot noise originates from the fluctuation in the electrical signal due to the discreteness of electron charges. In addition, it is well known that shot noise is useful to obtain information of transmission that is not available through the standard conductance measurements [35,36]. The shot noise is characterized by the Fano factor F being the ratio of the noise power to mean current [37]. Recently, some papers focus on the shot noise in semiconductor
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superlattices and graphene based nano-structures both theoretically and experimentally [7,8,38-51]. Cheianov and Falko studied the unusual Fano factor in the graphene n–p junctions [38]. Tworzydlo et al. predicted that the Fano factor for a wide and short sample has a maximum value of 1/3 at the Dirac point, which is 1/3 of Poissonian value [39]. This is the same value as in the diffusive metals. Zhu and Guo investigated shot noise in the graphenebased double barriers and found that the shot noise with the Fano factor equal to 1/3 occurs at the Dirac point [40]. The transport properties and shot noise in Thue-Morse sequence graphene superlattice were investigated by Huang et al. [42] the authors found that the Fano factor has a maximum close to 1/3 in the vicinity of Dirac point. Also, experimental results of shot noise measurement in graphene structures are in good agreement with the theoretical predication [7,52]. It is generally accepted that the sub-Poisson shot noise of graphene originated from the peculiar band structure at the low energy regime near the Dirac point. In other words, the quasiparticles in graphene are described by the massless Dirac-like equation rather than the Schröedinger equation. The purpose of this paper is to study the shot noise in magnetic field modulated graphene superlattice by using the transfer matrix method. The effect of the number of barriers on the Fano factor is taken into account. We show that for the antiparallel magnetization configuration or when magnetic field is zero the Fano factor reaches its maximum value 1/3, whereas for the parallel magnetization configuration it can be approached to 1 due to the transmission blockage. In particular, our probe shows that for the parallel magnetization and when the number of barriers, N, are larger than 50 the Fano factor reaches a Poissonian value, which is dramatically distinguishable from the semiconductor superlattice. In the semiconductor superlattice when all the barriers are identical, in the N → ∞ limit the shot noise approach 1/3 of the Poissonian value [50,51]. The rest of the paper is organized as
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follows; our method and formalism are described in the next section. In Section 3 we present and discuss our results, and finally we end the paper with a brief conclusion.
2. Model and theory In this paper, we consider two kinds of systems. In both cases, a monolayer graphene covered by a thin insulating layer and parallel ferromagnetic (FM) stripes are deposited on the top of the dielectric layer [25,26]. Further, a gate voltage is applied to the FM stripes in order to produce an electrostatic barrier. In both cases, FM stripes have magnetization perpendicular to the graphene in the x-y plane. In the first and the second cases FM stripes with magnetization parallel (P) or antiparallel (AP) perpendicular to the graphene in the x-y plane are deposited on top of the dielectric layer, respectively. As known, the magnetic field generated by such a strip is non-uniform and depends on both its size and the distance to the studied 2D system. However, for sufficiently thin ferromagnetic strip, it is almost constant under the strip and vanishes outside this region [53]. Thus, the systems under consideration are graphene superlattice modulated by magnetic field. The magnetic field B(x) emerging from the FM stripes will influence locally the motion of Dirac electrons in the graphene x-y plane and is assumed to be homogeneous in the y direction, but varies along the x direction. The schematic of the structures is shown in Fig. 1. According to the above discussion, the potential profile of the system consists of a sequence of N electrostatic barriers of equal height U 0 and width b, modulated by N magnetic barriers. Where, electrostatic and magnetic barriers separated by well regions (nonmagnetic regions) of width w. Here, the effects of electron-electron and electron-phonon interactions are neglected by considering a single electron transmission at zero temperature. Therefore, the charge carriers in our model are described by the following Hamiltonian
Hˆ Hˆ 0 V ( x) Iˆ,
(1)
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in which, ˆ v σ H .( p eA) , 0 F
(2)
U V ( x) 0 0
(3)
in barrier, in well,
, where p ( px , px ) is the quasiparticle momentum (setting =1), σ ( x , x ) is the 2D Pauli matrix, vF 106 ms 1 is the Fermi velocity, Iˆ is a 2×2 unit matrix, and A =[0, Ay (x) ,
0] is the vector potential with x A( x) B( x) in the Laudau gauge. For simplicity, we express all the quantities in the dimensionless form by rescaling: the magnetic field B( x) B0 B( x), the vector potential
A( x) B0l B A( x) with the magnetic length l B / eB0 , the
wavevector k k / lB , the electron energy E EvF / lB , the barrier height U 0 U 0 vF / lB , the barrier of width b bl B and well of with w wl B . For a realistic value B0 0.1 T [54,55] we find l B 80 nm and vF / lB 7 meV, which set the typical length and energy scales. To solve Eq. (1), we suppose that an incident electron from the left will go towards the interface with incident angle and energy E. Moreover, for the sake of simplicity we divide barrier regions into M (M 1) segments, each of which has width a =b/M. The vector potential in each segment can be considered as constant and then the plane-wave
ˆ can be approximation can be taken [56]. The general solution to the Hamiltonian H expressed in the following form
b e
ik y y
1 a e iqx x q x ik y i E U0
1 iq x x q ik y bi e x E U0
,
1 d e ik x x k x ik y i E
,
(4)
in barriers and
w e
ik y y
1 ci e ik x x k x ik y E
5
(5)
in wells. Where E is the energy of incident electron, q x ( E U 0 ) 2 k y2 and k x E cos are the wave vectors along the x direction in the barrier and well regions respectively, while k y E sin A( x) is the wave vectors along the y axis. Also a, b and c, d represent the
ampliudes of quasiparticles in the barrier and well regions, respectively. By appling the continuity of wave functions at the boundaries for a system consisting of N barriers and using the transfer-matrix method, we obtain t and r which represent the reflection and transmission coefficients, respectively. Then transmission probability T and reflection probability R can be evaluated by T P ( AP ) t P ( AP )
2
and R P ( AP ) r P ( AP )
2
respectively. Here, P
and AP denote the parallel and antiparallel alignments, respectively. Thus, the zero temperature conductivity through the magnetic field modulated graphene superlattice can be calculated by means of Büttiker formula [57] /2
G P ( AP ) G0
T
P ( Ap )
( ) cos( )d ,
(6)
/2
where G0 2e 2 EF Ly / vF 2 in which L y denotes the width of the graphene strip along the y direction. A convenient measure of shot noise is the Fano factor, and it was defined as the ratio of noise power and mean current and given by [39] /2
F P ( AP )
T
/2
P ( Ap )
(1 T P ( AP ) )( ) cos( )d
T
(7)
.
/2
P ( AP )
( ) cos( )d
/2
According to the Eq. (7) the Fano factor F=0 in the perfect transmission case and F=1 when the transmission is blocked.
3. Numerical result and discussion
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In this section we investigate the transmission probability and shot noise in the magnetic field modulated monolayer graphene superlattice by using the methods described in the previous section. In all the calculations, energy E of the incident electrons, barrier height U 0 , well width w, barrier width b and the magnetic field B are taken to be 5, 10, 0.2, 0.2 and 1 respectively, otherwise it is specified. At first, we calculate the transmission probability as a function of barrier height and incident angle through the system with different number of barriers when magnetic field is zero. The results are shown in Fig. 2. As it is clear from Fig. 2 transmission probability is very small at the U 0 E (1 w / b) . Thus, this point is new Dirac-like point. This phenomenon is very clear when the number of barriers is large enough. The new Dirac-like point is determined by the ratio of w/b. By using the transfer-matrix method, one can find the expression for the transmission probability T ( ) t at the new Dirac-like point as
t
(2 cos ) 2 N 1 , i ( 2 k xl2 N ) 2 1 4[ M 11 sin i sin ( M 12e M 21e i ) ) M 22ei 2 k xl2 N ]
M 11 M 12 2 N 1 2i sin i 2ik l M 21 M 22 i 2 2e e x i
2e i e 2ik xli , 2i sin
(8)
(9)
where li is the length of system at ith boundary. In addition, due to the interface of the well and the barrier, the transmission probability oscillates with the variation of the potential barriers height and by increasing the number of barriers more resonant peaks (valleys) appear in the transmission probability at some certain U 0 . The resonance condition is given by a function that yields f (q x , w, b) . For instance, for
the case N=1, the condition yields q x b m (m is an integer). For N 2 when we have the condition that (q x k x ) (b w) 2m , the transmission has finite values at angles different from 0 . This is due to the resonance process in a system with N barriers [19]. Fig. 3 shows the Fano factor with different number of barriers when magnetic field is zero. In the 7
vicinity of the Dirac-like point, the Fano factor shows a peak with value approximately F=1/3 which is originating from quasiparticle chirality and Klein tunneling. Meanwhile, in the region far from this point it oscillates with decreasing amplitude. This is a result of the completely ballistic tunneling, since the classical dynamics of the Dirac fermions is ballistic [39]. The Fano factor is sensitive to the ratio of w/b, when the well and the barrier width are changed simultaneously, the Fano factor shows the robust properties, because the position of the new Dirac point does not shift. With increasing the well width, the peak of the Fano factor shift right. It should be noted that if we choose all the parameters as Ref. [58] then figure 3 would be similar to figure 3 in Ref. [58]. With the condition that the magnetic field is applied and the antiparallel magnetization configuration, the Fano factor was calculated and the results was plotted in Fig. 4. As it is clear in Fig. 4, the Fano factor is a maximum F = 1/3 near the U 0 E (1 w / b) , while it oscillates with smaller amplitude in the region far from this point. This is similar to the zero magnetic field case, as a result when a monolayer graphene superlattice modulated by N FM stripes with antiparallel magnetization the position of the new Dirac point is robust against the magnetic field. It is necessary to mention that in Fig. 4, B=1 and the other parameters in Figs. 4(a), (b) and (c) are the same as in Figs. 3(a), (b) and (c), respectively. Fig. 5 presents the Fano factor in magnetic field modulated monolayer graphene superlattice. In this figure FM stripes with magnetization parallel is perpendicular to the graphene layer. As it is obvious in Fig. 5 the Fano factor increases by increasing the number of barriers. We note that when the number of barriers is big enough unlike antiparallel magnetization structure the Fano factor reaches a Poissonian value. Because the transmission can be blocked by the magnetic-electric barrier provided by the ferromagnets. These features can be intuitively conceived from the following discussion. The emergence angle f at right is obtained by exploiting conservation of wave vector in y direction 8
(8)
(10)
sin f sin Nb / E.
Eq. (10) implies that transmission through the structure is only possible when satisfied the following condition 1 sin
Nb 1. E
(11)
Another explanation for this phenomenon is that when sin ( Nb / E ) 1 the electron wave vector in the 2Nth strip is imaginary, which corresponds to an evanescent wave. Thus the transmission probability decreases by increasing N and/or decreasing E. In other words, by controlling the ratio Nb / E one can control the Fano factor and this renders the structure’s efficient wave-vector filter for Dirac electrons. It is observed from Fig. 5c that a striking Poisson value plateau of the Fano factor is formed for any barrier height. Because for N 50 the sin ( Nb / E ) is always more than 1, which means for any barrier height the Fano factor is 1. To further understand the variation of the Fano factor in magnetic field modulated graphene superlattice, we calculated the Fano factor as a function of the number of barriers with w=b=0.2. The result is plotted in Fig. 6. We can see that at U 0 2 E for the antiparallel magnetization configuration or when magnetic field is zero, the Fano factor is far below 1/3. Moreover, at U 0 2 E and for one barrier structure the Fano factor is far away from 1/3. By increasing the number of barriers, a periodical structure is formed, the point U 0 2 E becomes a Dirac-like point, and the Fano factor in this point approaches 1/3. Meanwhile, from Fig. 6, we find that in the parallel magnetization configuration as the number of magnetic barriers increases, the shot noise increases. It is also very clear when the number of barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value for any barrier height due to the transmission blockage.
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4. Conclusion Based on the transfer-matrix method, we have investigated the shot noise in magnetic field modulated graphene superlattice. We have shown that for the antiparallel magnetization configuration or in absence of the magnetic field the Fano factor shows a peak at the new Dirac point with value approximately F=1/3. This is due to massless Dirac fermions and directly related to Klein tunneling. In other words, at the new Dirac point the transport is almost prohibited, as a result the conductivity gets the minimum value and the Fano factor reaches the maximum value. Meanwhile, for the antiparallel magnetization the position of the new Dirac point is robust against the magnetic field. Besides, for the parallel magnetization the Fano factor increases by increasing the number of barriers. When the number of barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value due to the transmission blocked by the magnetic-electric barrier provided by the ferromagnets.
References [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Griegorieva, A. A. Firsov, Science 306 (2004) 666. [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Griegorieva, A. A. Firsov Nature (London) 438 (2005) 197. [3] Y. Zhang, Y. Wen Tan, H. L. Stormer, P. Kim, Nature 438 (2005) 201. [4] K. Ziegler, Phys. Rev. B 75 (2007) 233407. [5] M. I. Katsnelson, Eur. Phys. J. B 51 (2006) 157. [6] M. I. Katsnelson, K. S. Nososelov, A. K. Geim, Nat. phys. 2 (2006) 620. [7] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, P. J. Hakonen, Phys. Rev. Lett. 100 (2008) 196802.
10
[8] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Mor purgo, P. J. Hakonen, J. Low Temp. Phys. 153 (2008) 374. [9] C. W. J. Beenakker, Phys. Rev. Lett. 97 (2006) 067007. [10] L. Esaki, R. Tsu and, IBM J. Res. Develop. 14 (1970) 61. [11] F. Z. Meghoufel, S. Bentata, S. Terkhi, F. Bendahma, S. Cherid, Superlattices Microstruct 57 (2013) 115. [12] G. Bastard, E. E. Mendez, L. L. Chang, L. Esak, Phys. Rev. B 28 (1983) 3241. [13] A. Sibille, J. F. Palmier, h. Wang, F. Mollot, Phys. Rev. Lett. 64 (1990) 52. [14] James Leo, Phys. Rev. B 39 (1989) 5947. [15] J. C. Meyer, C. O. Girit, M. F. Crommie, A. Zettl, Appl. Phys. Lett. 92 (2008) 123110. [16] S. Marchini, S. Günther, J. Wintterlin, Phys. Rev. B 76 (2007) 075429. [17] A. L. Vazquez de Parga, F. Calleja, B. Borca, M. C. G. Passeggi, Jr., J. J. Hinarejos, F. Guinea, R. Miranda, Phys. Rev. Lett. 100 (2008) 056807. [18] C. Bai, X. Zhang, Phys. Rev. B 76 (2007) 075430. [19] N. Abedpour, A. Esmailpour, R. Asgari, M. R. Rahimi Tabar, Phys. Rev. B 79 (2009) 165412. [20] F. Sattari, E. Faizabadi, Solid State Commun 79 (2014) 48. [21] E. Faizabadi, M. Esmaeilzadeh, F. Sattari, Eur. Phys. J. B 85 (2012) 30073. [22] L. Brey, H. A. Fertig, Phys. Rev. Lett. 103 (2009) 046809. [23] C. H. Park, L. Yang, Y. W. Son, M. L. Cohen, S. G. Louie, Phys. Rev. Lett. 101 (2008) 126804. [24] A. De Martino, L. Dell’Anna, R. Egger, Phys. Rev. Lett. 98 (2007) 066802. [25] B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, D. Goldhaber-Gordon, Phys. Rev. Lett. 98 (2007) 236803. [26] J. R. Williams, L. Di Carlo, C. M. Marcus, Science 317 (2007) 638. [27] L. Dell’Anna, A. De Martino, Phys. Rev. B 79 (2009) 045420. 11
[28] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, J. Phys.: Condens. Matter 22 (2009) 465302. [29] L. Z. Tan, C.-H. Park, S. G. Louie, Phys. Rev. B 81 (2010) 195426. [30] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, Appl. Phys. Lett. 93 (2008) 242103. [31] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, New J. Phys. 11 (2009) 095009. [32] Q. Sh. Wu, Sh. N. Zhang, Sh. J. Yang, J. Phys.: Condens. Matter 20 (2008) 485210. [33] Q. H. Huo, R. Z. Wang, H. Yan, Appl. Phys. Lett. 101 (2012) 152404. [34] F. Sattari, E. Faizabadi, Eur. Phys. J. B 86 (2013) 40275. [35] C. W. J. Beenakker, Phys. Rev. B 46 (1992) 1889. [36] J. C. Egues, G. Burkard, D. S. Saraga, J. Schliemann, D. Loss, Phys. Rev. B 72 (2005) 235326. [37] Y. M. Blanter, M. Büttiker, Phys. Rep. 336 (2000) 1. [38] V. V. Cheianov, V. I. Falko, Phys. Rev. B 74 (2006) 041403. [39] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, C. W. J. Beenakker, Phys. Rev. Lett. 96 (2006) 246802. [40] R. Zhu, Y. Guo, Appl. Phys. Lett. 91 (2007) 252113. [41] Y .Gong, Y.Guo, J. Appl. Phys. 106 (2009) 064317. [42] H. Huang, D. Liu, H. Zhang, X. Kong, J. Appl. Phys. 113 (2013) 043702. [43] E. B. Sonin Phys. Rev. B 77 (2008) 233408. [44] Y-Xian Li, L.-Fei. Xu, Solid State Commun 151 (2011) 219. [45] E. S. Azarova, G. M. Maksimova, Physica E 61 (2014) 118. [46] Q. P. Wu, X. D. He, Z. F. Liu Eur. Phys. J. B 85 (2012) 346. [47] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, P. J. Hakonen, Solid State Commun 149 (2009) 1050. [48] S. Wu, Y.Guo, Physica E 59 (2014) 158. [49] B. H. Wu, J.C. Cao, Physica E 40 (2008)1319. 12
[50] M. J. M. de Jong, C. W. J. Beenakker, Phys. Rev. B 51 (1995) 16867. [51] W. Song, A. K. M. Newaz, J. K. Son, E. E. Mendez, Phys. Rev. Lett. 96 (2006) 126803. [52] L. DiCarlo, J. R. Williams, Y. Zhang, D. T. McClure, C. M. Marcus, Phys. Rev. Lett. 100 (2008) 156801. [53] A. Matulis, F. M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 1518. [54] F. Zhai, K. Chang, Phys. Rev. B 77 (2008) 113409. [55] A. T. Ngo, J. M. Villas-Boas, S. E. Ulloa, Phys. Rev. B 78 (2008) 245310. [56] Z. Y. Zeng, L. D. Zhang, X. H. Yan, J. Q. You, Phys. Rev. B 60 (1999) 1515. [57] M. Büttiker, Phys. Rev. Lett. 57 (1986) 1761. [58] X. X. Guo, D. Liu, Y. X. Li, Appl. Phys. Lett. 98 (2011) 242101.
Figure captions: Fig. 1. (a) Schematic diagram of the model, the monolayer graphene covered by a thin insulating layer, parallel FM stripes are deposited on top of the insulating layer. Each FM stripe has a rectangular cross section and a magnetization parallel to the z axis. The gate voltage V g applied on the FM stripes induces potential barrier in the graphene sheet. The FM stripes have magnetization parallel (P) or antiparallel (AP) to the z axis. (b) Magnetic field profile B(x) (red dashed line), corresponding vector potential A(x) (blue solid line) and the electrostatic potential U 0 (green dotted line) for the P alignment. (c) The same as in (b) but for the AP alignment.
Fig. 2. The transmission probability as a function of barrier height and incident angle for electrons traversing the considered structure. The number of barriers is (a) five, (b) twenty and [(c) and (d)] fifty. The well width is [(a), (b) and (c)] 0.2 and (d) 0.3. Fig. 3. The Fano factor as a function of barrier height for graphene superlattice with different the number of barriers in absence of the magnetic field. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
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Fig. 4. The Fano factor as a function of barrier height for graphene superlattice modulated by N FM stripes with antiparallel magnetization. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
Fig. 5. The Fano factor as a function of barrier height for graphene superlattice modulated by N FM stripes with parallel magnetization configuration. The number of barriers is (a) five, (b) twenty and (c) fifty. The red solid line, blue dashed line and black dotted line correspond to w=0.2, w=0.3 and w=0.4, respectively.
Fig. 6. The Fano factor as a function of the number of barriers for graphene superlattice modulated by N FM stripes with antiparallel configuration magnetization or when magnetic field is zero (green dashed and blue solid line) and parallel magnetization configuration (red dotted and black dashed-dotted line). Green dashed and black dashed-dotted line correspond to U 0 2E and blue solid and red dotted line correspond to U 0 2E .
Fig. 1.
Fig. 2.
14
Fig. 3.
15
Fig. 4.
Fig. 5.
16
Fig. 6.
Graphical abstract In the parallel magnetization configuration when the number of the barriers is large enough unlike untiparallel magnetization configuration the Fano factor reaches a Poissonian value for any barrier height.
Highlights We investigate the shot noise properties through graphene superlattice.
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The Fano factor depends on the number of barriers. For the parallel configuration the Fano factor reaches a Poissonian value. For the antiparallel configuration the Fano factor at the Dirac point equals 1/3.
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